We show that the patterns in the Abelian sandpile are stable. The proof combines the structure theory for the patterns with the regularity machinery for non-divergence form elliptic equations. The stability results allow one to improve weak-\(*\) convergence of the Abelian sandpile to pattern convergence for certain classes of solutions.

中文翻译：

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阿贝尔沙堆中图案的稳定性

我们表明，阿贝尔沙堆中的模式是稳定的。该证明将模式的结构理论与非散度形式椭圆方程的正则性机制相结合。稳定性结果使人们可以改善Abelian沙堆的弱\（* \）收敛，从而对某些类的解决方案进行模式收敛。